Working with the imaginary number 'i' seems complex, but simplifying them follows a few simple, beautiful rules.

The Most Important Rule: i² = -1. This is the definition. It is the key to all simplification.

When adding or subtracting complex numbers, you just combine like terms. Add the real parts together and the imaginary parts together.

Example: (3 + 2i) + (5 - 4i) becomes (3+5) + (2i - 4i), which simplifies to 8 - 2i.

When multiplying, you treat it just like multiplying binomials (FOIL method).

Example: (2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i².

Now, use the magic rule! The '15i²' becomes 15(-1), which is -15.

Finally, combine like terms: (8 - 15) + (10i + 12i) = -7 + 22i.

The key is to always remember that 'i' is not just a variable. It has a property. i² simplifies to -1.

Follow these simple rules, and the world of complex numbers becomes surprisingly simple to navigate.